δ&ApplyFunction;a&plus;b&equals;δ&ApplyFunction;a&plus;δ&ApplyFunction;b for any a and b in k

2.

δ&ApplyFunction;ab&equals;σ&ApplyFunction;a&InvisibleTimes;δ&ApplyFunction;b&plus;δ&ApplyFunction;a&InvisibleTimes;b for any a and b in k

Example: For any alpha in k, delta[alpha] is defined as alpha(sigma - 1). The map alpha[delta] given by delta[alpha]a = alpha(sigma(a) - a) is called an inner derivation.

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Lemma 1. Let k be a field, sigma be an automorphism of k, and delta be a pseudo-derivation of k.

3.

If sigma <> 1, then there is an element alpha in k such that:

δ&equals;α&InvisibleTimes;σ−1&InvisibleTimes;` = `&InvisibleTimes;δα

4.

If delta <> 0, then there is an element beta in k such that:

σ&equals;β&InvisibleTimes;δ&plus;1

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Definition 2. (Univariate skew-polynomials) The left skew polynomial ring given by sigma and delta is the ring (k[x], +, .) of polynomials in x over k with the usual polynomial addition, and the multiplication given by:

To distinguish it from the usual commutative polynomial ring k[x], the left skew polynomial ring is denoted by k[x; sigma, delta]. Its elements are called skew polynomials or Ore polynomials. It can be shown that k[x; sigma, delta] possesses the right and left Euclidean division algorithms.

Note: The product x^i a[i] must be computed in the ring k[x; sigma*, delta*]. It is easy to show that (sigma*)* = sigma, (delta*)* = delta. You can also verify that that the adjoint is a linear bijective map and that (M o N){*} = N* o M*.

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Lemma 4. Let theta be a pseudo-linear map with respect to sigma and delta. Then: